Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,10)(6,11)(7,14)(12,13)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,41)(36,40)(37,39)(38,42)(43,47)(44,46)(45,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(18,20)(19,26)(25,28)(27,29)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,4,6,10)(2,8,11,5)(3,12,14,16)(7,15,9,13)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,9)(4,12)(5,15)(6,14)(7,11)(8,13)(10,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5,6,8)(2,10,11,4)(3,13,14,15)(7,16,9,12)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9633557);